Enriques Surfaces I

This is the first of two volumes representing the current state of knowledge about Enriques surfaces which occupy one of the classes in the classification of algebraic surfaces. Recent improvements in our understanding of algebraic surfaces over fields of positive characteristic allowed us to approach the subject from a completely geometric point of view although heavily relying on algebraic methods. Some of the techniques presented in this book can be applied to the study of algebraic surfaces of other types. We hope that it will make this book of particular interest to a wider range of research mathematicians and graduate students. Acknowledgements. The undertaking of this project was made possible by the support of several institutions. Our mutual cooperation began at the University of Warwick and the Max Planck Institute of Mathematics in 1982/83. Most of the work in this volume was done during the visit of the first author at the University of Michigan in 1984-1986. The second author was supported during all these years by grants from the National Science Foundation.

0. Preliminaries. - S1. Double covers. - S2. Rational double points. - S3. Del Pezzo surfaces. - S4. Symmetric quartic Del Pezzo surfaces. - S5. Symmetric cubic Del Pezzo surfaces. - S6. Prym canonical maps. - S7. The Picard scheme. - Bibliographical notes. - I. Enriques surfaces: generalities. - S1. Classification of algebraic surfaces. - S2. The Picard group. - S3. The K3-cover. - S4. Differential invariants. - S5. Riemann-Roch and a vanishing theorem. - S6. Examples. - Bibliographical notes. - II. Lattices and root bases. - S1. Generalities. - S2. Root bases and their Weyl groups. - S3. Root bases of finite and affine type. - S4. Root bases of hyperbolic type. - S5. The Enriques lattice. - S6. The Reye lattice. - S7. The function ? M. - S8. 2-congruence subgroups of finite Weyl groups. - S9. The factor group W/W(2). - S10. The structure of W(2). - Bibliographical notes. - Tables. - III. The geometry of the Enriques lattice. . - S1. Divisors of canonical type. - S2. The nodal chamber. - S3. Canonical r-sequences and U[r]-markings. - S4. U-markings. - S5. U[3]-markings. - S6. Linear systems |C| with C2 ? 10. - Bibliographical notes. - IV. Projective models. . - S1. Preliminaries. - S2. Linear systems on K3-surfaces. - S3. Numerical connectedness. - S4. Base-points. - S5. Hyperelliptic maps. - S6. Birational maps. - S7. Superelliptic maps. - S8. The branch locus of superelliptic maps. - S9. Projective models of degree ? 10. - S10. Applications to linear systems. - Appendix. A theorem of Igor Reider. - Bibliographical notes. - V. Genus 1 fibrations. . - S1. Genus 1 fibrations:generalities. - S2. The Picard group. - S3. Jacobian fibrations. - S4. Ogg-Shafarevich theory. - S5. Weierstrass models. - S6. Genus 1 fibrations on rational surfaces. - S7. Genus 1 fibrations on Enriques surfaces. - Bibliographical notes. - Glossary ofnotations.